Average Error: 42.2 → 22.8
Time: 52.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -6.812585307839581 \cdot 10^{+230}:\\ \;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\ \mathbf{elif}\;n \le -2.5419696905962584 \cdot 10^{+96}:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le -889817905.8903499:\\ \;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\ \mathbf{elif}\;n \le 1.658674653647369 \cdot 10^{-94}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -6.812585307839581 \cdot 10^{+230}:\\
\;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\

\mathbf{elif}\;n \le -2.5419696905962584 \cdot 10^{+96}:\\
\;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;n \le -889817905.8903499:\\
\;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\

\mathbf{elif}\;n \le 1.658674653647369 \cdot 10^{-94}:\\
\;\;\;\;0\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\

\end{array}
double f(double i, double n) {
        double r5030915 = 100.0;
        double r5030916 = 1.0;
        double r5030917 = i;
        double r5030918 = n;
        double r5030919 = r5030917 / r5030918;
        double r5030920 = r5030916 + r5030919;
        double r5030921 = pow(r5030920, r5030918);
        double r5030922 = r5030921 - r5030916;
        double r5030923 = r5030922 / r5030919;
        double r5030924 = r5030915 * r5030923;
        return r5030924;
}
↓
double f(double i, double n) {
        double r5030925 = n;
        double r5030926 = -6.812585307839581e+230;
        bool r5030927 = r5030925 <= r5030926;
        double r5030928 = 100.0;
        double r5030929 = i;
        double r5030930 = 0.5;
        double r5030931 = r5030925 * r5030930;
        double r5030932 = r5030929 * r5030931;
        double r5030933 = r5030929 * r5030929;
        double r5030934 = r5030925 * r5030933;
        double r5030935 = 0.16666666666666666;
        double r5030936 = r5030934 * r5030935;
        double r5030937 = r5030925 + r5030936;
        double r5030938 = r5030932 + r5030937;
        double r5030939 = r5030928 * r5030938;
        double r5030940 = -2.5419696905962584e+96;
        bool r5030941 = r5030925 <= r5030940;
        double r5030942 = 1.0;
        double r5030943 = r5030929 / r5030925;
        double r5030944 = r5030942 + r5030943;
        double r5030945 = pow(r5030944, r5030925);
        double r5030946 = r5030945 / r5030943;
        double r5030947 = /* ERROR: no posit support in C */;
        double r5030948 = /* ERROR: no posit support in C */;
        double r5030949 = r5030942 / r5030943;
        double r5030950 = r5030948 - r5030949;
        double r5030951 = r5030928 * r5030950;
        double r5030952 = -889817905.8903499;
        bool r5030953 = r5030925 <= r5030952;
        double r5030954 = 1.658674653647369e-94;
        bool r5030955 = r5030925 <= r5030954;
        double r5030956 = 0.0;
        double r5030957 = r5030955 ? r5030956 : r5030939;
        double r5030958 = r5030953 ? r5030939 : r5030957;
        double r5030959 = r5030941 ? r5030951 : r5030958;
        double r5030960 = r5030927 ? r5030939 : r5030959;
        return r5030960;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.2
Target42.0
Herbie22.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -6.812585307839581e+230 or -2.5419696905962584e+96 < n < -889817905.8903499 or 1.658674653647369e-94 < n

    1. Initial program 53.5

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied div-sub53.5

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)}\]

    4. Taylor expanded around 0 20.3

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right) + \left(n + \frac{1}{2} \cdot \left(i \cdot n\right)\right)\right)}\]

    5. Simplified20.3

      \[\leadsto 100 \cdot \color{blue}{\left(\left(n + \frac{1}{6} \cdot \left(n \cdot \left(i \cdot i\right)\right)\right) + i \cdot \left(n \cdot \frac{1}{2}\right)\right)}\]

    if -6.812585307839581e+230 < n < -2.5419696905962584e+96

    1. Initial program 44.8

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied div-sub44.8

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)}\]
    4. Using strategy rm

    5. Applied insert-posit1636.0

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)\right)} - \frac{1}{\frac{i}{n}}\right)\]

    if -889817905.8903499 < n < 1.658674653647369e-94

    1. Initial program 27.2

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

    2. Taylor expanded around 0 21.2

      \[\leadsto \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification22.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.812585307839581 \cdot 10^{+230}:\\ \;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\ \mathbf{elif}\;n \le -2.5419696905962584 \cdot 10^{+96}:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le -889817905.8903499:\\ \;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\ \mathbf{elif}\;n \le 1.658674653647369 \cdot 10^{-94}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \left(n \cdot \frac{1}{2}\right) + \left(n + \left(n \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))